A reproducing kernel Hilbert space approach to singular local stochastic volatility McKean–Vlasov models

Christian Bayer (), Denis Belomestny (), Oleg Butkovsky () and John Schoenmakers ()
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Christian Bayer: Weierstrass Institute
Denis Belomestny: Duisburg-Essen University
Oleg Butkovsky: Weierstrass Institute
John Schoenmakers: Weierstrass Institute

Finance and Stochastics, 2024, vol. 28, issue 4, No 7, 1147-1178

Abstract: Abstract Motivated by the challenges related to the calibration of financial models, we consider the problem of numerically solving a singular McKean–Vlasov equation d X t = σ ( t , X t ) X t v t E [ v t | X t ] d W t , $$ d X_{t}= \sigma (t,X_{t}) X_{t} \frac{\sqrt{v}_{t}}{\sqrt{\mathbb{E}[v_{t}|X_{t}]}}dW_{t}, $$ where W $W$ is a Brownian motion and v $v$ is an adapted diffusion process. This equation can be considered as a singular local stochastic volatility model. While such models are quite popular among practitioners, its well-posedness has unfortunately not yet been fully understood and in general is possibly not guaranteed at all. We develop a novel regularisation approach based on the reproducing kernel Hilbert space (RKHS) technique and show that the regularised model is well posed. Furthermore, we prove propagation of chaos. We demonstrate numerically that a thus regularised model is able to perfectly replicate option prices coming from typical local volatility models. Our results are also applicable to more general McKean–Vlasov equations.

Keywords: Stochastic volatility models; Singular McKean–Vlasov equations; Reproducing kernel Hilbert space; 91G20; 65C30; 46E22 (search for similar items in EconPapers)
JEL-codes: C63 G17 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s00780-024-00541-5

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